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 "title": "Transit-timing variation",
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 "excerpt": "Transit-timing variation (TTV) is an exoplanet detection method that measures deviations from a linear ephemeris in transit times, caused by gravitational interactions between planets, strongest near mean-motion resonance.",
 "snippet": "Transit-timing variation (TTV) is an exoplanet detection method that measures deviations from a linear ephemeris in transit times, caused by gravitational interactions between planets, strongest near mean-motion resonance.",
 "node": "physical.astronomy.cosmos.cosmos.techniques",
 "markdown": "# Transit-timing variation\n\nTransit-timing variation (TTV) is a method that detects and characterizes exoplanets by measuring deviations from a linear ephemeris (table predicting an object's positions/times over time) in the times when a transiting planet occults its star. The deviations arise from gravitational interactions among the planets, which pull each orbit away from a perfect Keplerian period, and they are strongest when planets lie near a mean-motion resonance (orbits whose periods form a simple whole-number ratio).<sup>[1](https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staf334/8093570)</sup> Of the two techniques, TTVs give the higher equivalent mass precision overall.<sup>[2](https://arxiv.org/html/1706.09849)</sup>\n\n| Key fact | Value |\n|---|---|\n| What is measured | Residuals \\( O - C \\) of observed mid-transit times from a linear fit of period and epoch<sup>[3](https://ar5iv.labs.arxiv.org/html/astro-ph/0612442)</sup> |\n| Physical cause | Gravitational n-body interactions that make orbits deviate from Keplerian, strongest near mean-motion resonance<sup>[1](https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staf334/8093570)</sup> |\n| TTV period near resonance | \\( P_{\\mathrm{TTV}} = 1/\\lvert j/P_{2} - k/P_{1} \\rvert \\) for period ratio \\( P_{2}/P_{1} \\) within a few percent of \\( j/k \\)<sup>[2](https://arxiv.org/html/1706.09849)</sup> |\n| Chopping period | \\( P_{\\mathrm{syn}} = (1/P_{1} - 1/P_{2})^{-1} \\), the time between conjunctions<sup>[2](https://arxiv.org/html/1706.09849)</sup> |\n| Kepler-9 signal | Periods of 19.2 and 38.9 days varying at average rates of 4 and 39 minutes per orbit<sup>[4](https://www.science.org/doi/10.1126/science.1195778)</sup> |\n| Largest Kepler TTV | Kepler-88 b, amplitude about 12 hours<sup>[5](https://arxiv.org/abs/2609.24645)</sup> |\n| Precision compared with RV | No TTV mass better than 6 cm/s equivalent published since 2021; best radial-velocity precisions are 10–20 cm/s<sup>[2](https://arxiv.org/html/1706.09849)</sup> |\n\n## How it works\n\nTwo planets that gravitationally interact do not orbit with fixed periods. Each conjunction kicks the planets along their orbits, and when the period ratio \\( P_{2}/P_{1} \\) sits within a few percent of a ratio of small integers \\( j/k \\), these kicks accumulate coherently. The transit times then drift sinusoidally with the super-period\n\n\\[ P_{\\mathrm{TTV}} = 1/\\lvert j/P_{2} - k/P_{1} \\rvert, \\]\n\nwhich is set by the period ratio alone and is independent of mass.<sup>[2](https://arxiv.org/html/1706.09849)</sup> The amplitude of this resonant signal depends to first order on the perturber's period, the planet-to-star mass ratio, the perturber's eccentricity, and a function of the semi-major axis ratio.<sup>[5](https://arxiv.org/abs/2609.24645)</sup> Its strength depends on the planetary eccentricities raised to a power of the resonance order minus 1, where the order is \\( \\lvert j - k \\rvert \\); first-order resonances therefore dominate.<sup>[2](https://arxiv.org/html/1706.09849)</sup> In resonance, the signal also contains harmonics of the libration period and the apsidal precession period, the latter about 5 times longer; the libration period scales with \\( (m/M_{\\star})^{-2/3} \\), so unlike the super-period it does constrain the masses.<sup>[6](https://astro.mff.cuni.cz/davok/papers/ttv3_ApJ2016.pdf)</sup>\n\nA second, distinct signal arises at the synodic period,\n\n\\[ P_{\\mathrm{syn}} = (1/P_{1} - 1/P_{2})^{-1}, \\]\n\nthe time between successive conjunctions. This short-timescale component, called \"chopping\", has an amplitude set by the perturber's mass-to-star ratio and the semi-major axis ratio, and it can break the mass–eccentricity degeneracy of the resonant signal.<sup>[2](https://arxiv.org/html/1706.09849)</sup> For resonant systems the accumulated variation can reach order \\( (m_{\\mathrm{p}}/m_{\\mathrm{T}}) \\cdot P_{\\mathrm{T}} \\), nearly 15 minutes for an Earth-mass perturber of a Jupiter-mass transiting planet in a 3-day orbit.<sup>[3](https://ar5iv.labs.arxiv.org/html/astro-ph/0612442)</sup>\n\n## How it is done\n\nThe pipeline starts from the observed mid-transit times \\( t(j) \\), \\( j = 1 \\ldots N \\). Each transit light curve is fitted individually for a mid-transit time; the orbital period \\( P_{1} \\) is then estimated by linear regression, and the TTV is the residual \\( \\Delta t_j = t_j - (T_0 + E_j P_{1}) \\), where \\( T_0 \\) is the fitted epoch and \\( E_j \\) is the transit number relative to that epoch.<sup>[7](https://crimson.oca.eu/images/LAGRANGE/pages_perso/morby/papers/TTV.pdf)</sup> [Consistency](https://www.edgepedia.org/consistency) with a linear ephemeris is tested with a chi-square criterion.<sup>[2](https://arxiv.org/html/1706.09849)</sup> Because the resonant signal is sinusoidal on the super-period, the observing baseline should cover at least two super-periods, well sampled.<sup>[2](https://arxiv.org/html/1706.09849)</sup>\n\nInferring masses requires dynamical modeling of the \\( \\Delta t(j) \\) series. A central obstacle is the mass–eccentricity degeneracy: the same TTV curve can be produced by different combinations of perturber mass and eccentricity, because sampling at the transiting planet's period aliases short-period variations with the super-period.<sup>[5](https://arxiv.org/abs/2609.24645)</sup> Chopping terms can break this degeneracy.<sup>[2](https://arxiv.org/html/1706.09849)</sup>\n\n## Origin\n\nMean-motion-resonance TTVs were shown to reveal Earth-like planets through Solar-system-like perturbations, and TTVs were defined as the observable accumulation of transit period changes \\( O - C \\), with surveys estimated to measure transit intervals to 0.1–100 minutes for hundreds of jovian-mass and tens of terrestrial-mass planets.<sup>[8](https://www.science.org/doi/10.1126/science.1107822)</sup><sup> • </sup><sup>[2](https://arxiv.org/html/1706.09849)</sup> By 2007 the theory had been applied to specific systems, TrES-1 and HD 209458b, jointly with radial-velocity data to constrain any second planet.<sup>[9](https://vmnexsci.ipac.caltech.edu/workshop/2007/Holman.pdf)</sup><sup> • </sup><sup>[3](https://ar5iv.labs.arxiv.org/html/astro-ph/0612442)</sup> Lithwick, Xie, and Wu analyzed the super-period formula and the mass–eccentricity degeneracy in The Astrophysical Journal in 2012.<sup>[10](https://doi.org/10.1088/0004-637x/761/2/122)</sup> The first convincing detection came from Kepler: [Matthew Holman](https://www.edgepedia.org/matthew-j-holman) and colleagues reported in Science in 2010 two Saturn-size planets transiting [Kepler-9](https://www.edgepedia.org/kepler-9), confirmed by their timing variations.<sup>[11](https://doi.org/10.1126/science.1195778)</sup>\n\n## Variants\n\nTransit-duration variation (TDV) is the companion technique in which transit durations, rather than times, change. Duration changes arise from variations in semi-major axis, eccentricity, argument of periastron, or reorientation of the orbital plane, where inclination changes alter the transit chord length.<sup>[2](https://arxiv.org/html/1706.09849)</sup> Chopping constitutes a second signal mode, distinguished from the resonant super-period signal by its synodic timescale and its direct dependence on the perturber's mass ratio.<sup>[5](https://arxiv.org/abs/2609.24645)</sup> TTVs can also expose planets that never transit: a TTV signal induced by a non-transiting perturber was observed for [Kepler-19](https://www.edgepedia.org/kepler-19) b,<sup>[5](https://arxiv.org/abs/2609.24645)</sup> and [David Nesvorný](https://www.edgepedia.org/david-nesvorny) and colleagues detected and characterized a nontransiting planet by TTVs in the [KOI-142](https://www.edgepedia.org/koi-142) system, published in Science in 2012.<sup>[12](https://doi.org/10.1126/science.1221141)</sup>\n\n## Applications\n\nKepler-9 established the method's potential: its 19.2- and 38.9-day periods vary at average rates of 4 and 39 minutes per orbit, signatures of interaction near a 2:1 resonance, and the paper began a series of Kepler TTV discoveries that now includes more than 375 planets displaying TTVs in the NASA Exoplanet Archive as of 12/5/2024.<sup>[4](https://www.science.org/doi/10.1126/science.1195778)</sup> Kepler-88 b (KOI-142) shows a TTV amplitude of about 12 hours, the largest of the Kepler mission, earning the nickname \"The King of TTV\"; its roughly 0.6-day sinusoid on an 11-day orbit pointed to a non-transiting perturber near 2:1 resonance, which Barros et al. (2014) confirmed by radial velocity as Kepler-88 c, and Weiss et al. (2020) later found Kepler-88 d, about 3 Jupiter masses on a roughly 1340-day orbit.<sup>[5](https://arxiv.org/abs/2609.24645)</sup>\n\n## Limitations and alternatives\n\nTTV mass measurement requires transiting planets and companions close enough that the TTVs exceed the timing noise; the amplitude scales with the orbital period and the masses of the other bodies.<sup>[2](https://arxiv.org/html/1706.09849)</sup> Stellar spots crossing the planet's path increase the timing uncertainty of individual transits.<sup>[2](https://arxiv.org/html/1706.09849)</sup> The main structural limitation is the degeneracy problem: TTV analyses can constrain planetary masses relative to the star, as well as orbital elements, but absolute planetary masses require knowledge of the stellar mass.<sup>[16](https://doi.org/10.1088/0004-637x/802/2/116)</sup> Independent analyses of Kepler-9 b and c converged to masses of 45.1 ± 1.5 and 31.0 ± 1.0 \\( M_{\\oplus} \\) (Dreizler & Ofir 2014) and 43.5 ± 0.6 and 29.8 ± 0.6 \\( M_{\\oplus} \\) (Borsato et al. 2014), yet numerous multimodal solutions spanning a large mass range exist behind those tight ratios.<sup>[13](https://iopscience.iop.org/article/10.3847/1538-4357/ae74c9)</sup>\n\nAgainst radial velocity, TTV masses correlate well overall, with one outlier, Kepler-89d/KOI 94.01, where the TTV mass is smaller by a factor of about 2 than the RV mass.<sup>[2](https://arxiv.org/html/1706.09849)</sup> TTVs demand extensive observation, and no measurement better than 6 cm/s equivalent has been published since 2021, though TTVs give the higher equivalent precision of the two techniques overall.<sup>[2](https://arxiv.org/html/1706.09849)</sup> The TESS era has broadened the census: a catalog of TTVs from the first five years of TESS data lists systems including [TOI-216](https://www.edgepedia.org/toi-216), TOI-2525, TOI-1130, and [TOI-4504](https://www.edgepedia.org/toi-4504),<sup>[14](https://beta.iopscience.iop.org/article/10.3847/1538-4365/ae7a34)</sup> a 2025 catalog of Kepler/K2 systems re-observed by TESS enables searches for long-term trends in transit times,<sup>[1](https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staf334/8093570)</sup> and a homogeneous TTV investigation of all TESS systems with a confirmed single-transiting planet aims to detect unseen companions and characterize their masses and orbits.<sup>[15](https://www.aanda.org/articles/aa/abs/2026/01/aa57512-25/aa57512-25.html)</sup>\n\n## References\n\n1. [Transit ephemerides and timing variations from Kepler and K2 to TESS (MNRAS, 2025)](https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staf334/8093570)\n2. [Transit Timing and Duration Variations for the Discovery and Characterization of Exoplanets in the TESS era (Agol & Fabrycky review)](https://arxiv.org/html/1706.09849)\n3. [Developments in Planet Detection using Transit Timing Variations (Steffen 2006/2007)](https://ar5iv.labs.arxiv.org/html/astro-ph/0612442)\n4. [Kepler-9: A System of Multiple Planets Transiting a Sun-Like Star, Confirmed by Timing Variations (Holman et al. 2010, Science)](https://www.science.org/doi/10.1126/science.1195778)\n5. [Transit timing variation (review chapter)](https://arxiv.org/abs/2609.24645)\n6. [Dynamics and Transit Variations of Resonant Exoplanets (ApJ 2016)](https://astro.mff.cuni.cz/davok/papers/ttv3_ApJ2016.pdf)\n7. [Mass and Orbit Determination from Transit Timing Variations of Exoplanets (Nesvorný & Morbidelli)](https://crimson.oca.eu/images/LAGRANGE/pages_perso/morby/papers/TTV.pdf)\n8. [The Use of Transit Timing to Detect Terrestrial-Mass Extrasolar Planets (Agol et al. 2005, Science)](https://www.science.org/doi/10.1126/science.1107822)\n9. [Transit Timing (Holman, 2007 NExScI workshop presentation)](https://vmnexsci.ipac.caltech.edu/workshop/2007/Holman.pdf)\n10. [Yoram Lithwick, Jiwei Xie, Yanqin Wu (2012). EXTRACTING PLANET MASS AND ECCENTRICITY FROM TTV DATA. The Astrophysical Journal.](https://doi.org/10.1088/0004-637x/761/2/122)\n11. [Matthew J. Holman and colleagues (2010). Kepler-9: A System of Multiple Planets Transiting a Sun-Like Star, Confirmed by Timing Variations. Science.](https://doi.org/10.1126/science.1195778)\n12. [David Nesvorný and colleagues (2012). The Detection and Characterization of a Nontransiting Planet by Transit Timing Variations. Science.](https://doi.org/10.1126/science.1221141)\n13. [The Illusory Precision of Transit Timing Variation Masses: Hidden Solutions Behind Kepler-9's Tight Mass Ratio (ApJ)](https://iopscience.iop.org/article/10.3847/1538-4357/ae74c9)\n14. [Transit-timing Variations in TESS: A Catalog from the First 5 yr (ApJS)](https://beta.iopscience.iop.org/article/10.3847/1538-4365/ae7a34)\n15. [A homogeneous transit-timing-variation investigation of all TESS systems with a confirmed single-transiting planet (A&A, 2026)](https://www.aanda.org/articles/aa/abs/2026/01/aa57512-25/aa57512-25.html)\n16. [doi.org](https://doi.org/10.1088/0004-637x/802/2/116)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Astronomy › Cosmology and observation › Observational techniques: astrometry, photometry, spectroscopy*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
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 "speakable": "Transit-timing variation is an exoplanet detection method that measures deviations from a linear ephemeris in transit times, caused by gravitational interactions between planets, strongest near mean-motion resonance."
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